In this section we explain how one can derive the Darboux–Halphen equations from the Gauss–Manin connection of a universal family of elliptic curves.
This has been taken from the references
[1]}, [2]}. The family of elliptic curves
\(E_t\colon \ y^2-4(x-t_1)(x-t_2)(x-t_3)=0,\qquad t\in {\mathbb {C}}^3\backslash \cup _{i,j}\lbrace t_i=t_j\rbrace ,\) 
